EXCESS DEMAND FUNCTIONS, EQUILIBRIUM PRICES

AND EXISTENCE OF EQUILIBRIUM

by

Kam-Chau Wong

Discussion Paper No. 283, October 1995

Center for Economic Research Department of Economics University of Minnesota Minneapolis, MN 55455

Excess Demand Functions, Equilibrium Prices, and Existence of Equilibrium

Kam-Chau Wong t

Department of Economics Chinese University of Hong Kong

Shatin, Hong Kong E-mail: kamchauwong@cuhk.hk

Phone: (852) 2609-7053 Fax: (852) 2603-5805

September 6, 1995

Abstract: For continuous excess demand functions, the existing literature

(e.g. Sonnenschein [1972, 1973], Mantel [1974], Debreu [1974], Mas-Colell

[1977], etc.) achieves a complete characterization only when the functions

are defined on special subsets of positive prices. In this paper, we allow the

functions to be defined on a larger class of price sets, (allowing, for exam­

ple the closed unit simplex, including its boundary). Besides characterizing

excess demands for a larger class of economies, it is also a useful tool for

proving other results. It allows us to characterize the equilibrium price set

for a larger class of economies. It also permits extending Uzawa's observa­

tion [1962]' by showing that Brouwer's Fixed-Point Theorem is implied by

the Arrow-Debreu Equilibrium Existence Theorem ([1954], Thm. I.).

JEL Classification: D51, Dl1, D50

Keywords: Excess demand, Exchange economy, Equilibrium, Fixed-point

t This paper was written while I was visiting the Department of Economics of

the University of Minnesota in the summer of 1995; the hospitality of the department

is gratefully acknowledged. I also wish to thank Professor M. Richter for his helpful

comments and suggestions on earlier versions of this paper.

1

1. Introduction

In the existing literature of characterizing continuous excess demand

functions (Sonnenschein [1972, 1973], Mantel [1974], Debreu [1974], Mas­

Colell [1977], etc.), a complete characterization is achieved only for the

special case where the functions are defined on compact subsets of posi­

tive prices.! By weakening the preference requirement of monotonicity to

insatiability, we are able to characterizing a larger class of excess demand

functions, even those defined at boundary points, where some prices are

zero. In particular, Theorem 1 below gives a complete characterization for

any continuous excess demand function defined on any compact subset of

nonnegative nonzero prices, including, e.g. the closed unit simplex. This

extension has applications in several areas. We give two examples.

The first application is a characterization of equilibrium price sets. In

the series of papers by Sonnenschein [1972, 1973] and Mas-Colell [1977], it

has been established that any non-empty compact set in the relative interior

of the closed unit simplex is indeed the set of equilibrium prices for some

exchange economy (with monotone preferences). By using Theorem 1, we

drop the "interior" res.triction, showing that any non-empty compact set

in the closed simplex is the set of equilibrium prices for some exchange

economy (with insatiable preferences).

The second application concerns the connection between two very fun­

damental theorems - Brouwer's Fixed-Point Theorem and the Arrow­

Debreu Equilibrium Existence Theorem ([1954], Thm. I). Uzawa [1962]

proved that Brouwer's Theorem is equivalent2 to the equilibrium existence

1 McFadden et aI. [1974] considered not-necessarily-continuous excess demand func­tions on the set of positive prices.

2 In a formal sense, all mathematical theorems are equivalent, since they can be derived from the same set of axioms for mathematics. But in the present context, we use "equivalent" in the usual informal sense, meaning that there is a fairly direct proof

2

theorem for excess demand functions defined on the closed unit simplex.

By Theorem 1, it follows that the latter is implied3 by the Arrow-Debreu

Equilibrium Existence Theorem for economies. Consequently, Brouwer's

Theorem is implied by the Arrow-Debreu Theorem. This formalizes De­

breu's observation4 that "the proof of existence of a competitive equilibrium

requires mathematical tools of the same power as a fixed point theorem"

(Debreu [1982], p. 720).

The proof of Theorem 1 also has some interest of its own. It uses

modifications of the methods of both McFadden et al. [1974] (for the de­

composition ofthe given function into individual excess demand functions)5

and Debreu [1974] (for the construction of preferences for the individual ex­

cess demand functions). It sheds some light on the complementary nature

of the basic ideas of these two different methods.

2. Statement of Results

Let6 R+ = {p E JRI : P ~ OJ, fl = {p E JR~ : E~=l Pi = I}, o fl= {p E fl : P » O}, S = {p E JRI : p » 0 & IIpll = I}, and Sf = {p E S :

Pi ~ f for every i} for every real number f > O.

We will consider any exchange economy E = {(b,wi,JR~)~=d such

of one from the other.

3 See Footnote 3 above.

4 Cf. also Sonnenschein [1973], p. 352 and Sonnenschein [1982], p. 691.

5 I.e. we use oblique projections, as given in McFadden et al. [1974], rather than the orthogonal projection of Debreu [1974]. The reason is that the former method allows us to handle situations in which a continuous excess function is defined at boundary points.

6 For any x, Y E R,I, we write x ~ Y to mean Xi ~ Yi for every i = 1,···, /, and write x> Y to mean Xi > Yi for every i = 1"", I. Also, ". " denotes the Euclidean norm.

3

that

for all i, Wi » 0 E JR~ and ti is a continuous, insa-tiable, strictly convex preference relation on the consump- (1) tion space JR~.

Let G ~ JR~ \ {O}. A function ( : G -+ JRl is an excess demand function if

( is continuous and such that:

WL) p. «(p) = 0 for every pEG,

H) «(p) = «(..\p) for every pEG and every ..\ > 0 with ..\p E G,

BB) «( G) + q » 0 for some q » 0 E JR1•

It is well-known that for any function ( : G -+ JR1, if there exists an

exchange economy E = {(b,wi,JR~)~=l} satisfying (1) and such that

I

{«(p)} = L: {z E JR~: [p. z ~ p. Wi] i=l (2)

I

& ('v'y E JR~)[(p. y ~ p. Wi) =* (z by)]} - L: {wd i=l

for all pEG, then ( is continuous and satisfies (WL), (H) and (BB), i.e.

( is an excess demand function. A partial converse is provided by the

Sonnenschein-Mantel-Debreu Theorem (Debreu [1974]), which shows that

for any f > 0, any excess demand function ( : Sf -+ JR1, there exists an

exchange economy C = {(b,Wi, JR~)~=d satisfying (1) and

every ti is monotone, (3)

and such that (2) holds for all p E Sf.7 By dropping (3), we obtain the

following result, which considers excess demand functions defined on any

compact subset of JR~ \{O} (including e.g. the whole closure of S, ~, etc.),

rather than just Sf.

7 Different variants of the theorem were obtained by McFadden et a!. [1974]. Mas­Colell [1977]. Mantel [1979] and others.

4

Theorem 1 (Characterization of Excess Demand Functions). Let

K be a compact subset of IR~ \{O}. Then for any ( : K -+ IRI to be an

excess demand function, it is necessary and sufficient that there exists an

exchange economy [ = {(ti, Wi, IR~)~=d satisfying (1) and such that (2)

holds for all p E K.8

Proof. The sufficiency is clear. And the necessity follows immediately

from Lemma 1 and Proposition 1 in Section 3. Q.E.D.

We will now discuss two applications of Theorem 1. The first one con­

cerns a characterization of equilibrium price sets. For any exchange econ­

omy [= {(b,Wi, lR~H=d, a competitive equilibrium is a tuple (p, (Xi)~=l)

such that p E 1R~, Xi E IR~ for all i = 1, .. " I, and

E.1) (2:!=1 Xi - 2:!=1 Wi) $ 0 and p. (2:!=1 Xi - 2:~=1 Wi) = 0,

E.2) (p. Xi $ P . Wi) & (Vy E IR~)[(p. y $ p. Wi) ~ (Xi ti y)]

for all i = 1, ... , I.

A competitive equilibrium price for [ is a vector p E IR~ such that

(p, (Xi)~=l) is a competitive equilibrium for [ for some vector (Xi)~=l' We

use E£ to denote the set of competitive equilibrium prices for the economy

[in ~.

Following the work of Sonnenschein [1972, 1973], Mas-Colell [1977]

found a necessary and sufficient condition for any set K in the relative o

interior ~ of ~ to be the set of equilibrium prices for an exchange economy

[ = {(b,Wi, IR~)~=l} satisfying (1) and (3), namely that K is a non­

empty compact set. In Theorem 2 below, we relax the requirement (3)

for economies, but we obtain a more general characterization condition for

equilibrium price sets by dispensing with the "interior" restriction.

8 By (H). it is clear that Theorem 1 also covers the case where K = R~ \{o}.

5

Theorem 2 (Characterization of Equilibrium Price Sets). A set

K ~ D. is a non-empty compact set if and only if K = Ee for some

exchange economy £ = {(ti,wi, JR~)~=d satisfying (1).

Proof. The "if' part is well-known. To show the "only if" part, it is clear

that if ( : D. -t JRI is an excess demand function and £ = {(ti, Wi, JR~ )I=d is an exchange economy such that (2) holds for every p E D., then we have:

Ec = E" where E, = {p ED.: ((p) ~ O}. By Theorem 1, it suffices to

construct an excess demand function ( : D. -t JRI with K = E,. To do this,

we will modify the methods given in Uzawa [1962, p. 61], in Mas-Colell

[1977, Corollary 1] and Mas-Colell [1985, p. 195]. First, we pick any p E K.

Then we define a function e : D. -t JR! by e(p) = p - (p . p/p . p)p. It is

clear that e is an excess demand function. It is also easily verified that for

every p E D., one has: e(p) ~ 0 if and only if p = p. Finally, we define a

function ( : D. -t JRI by ((p) = Ape(p), where Ap = minpeK lip - pli. Then

clearly the function ( is an excess demand such that E, = K, and we are

done. Q.E.D.

We now discuss our second application of Theorem 1. As e?lphasized

by Sonnenschein [1972, 1973] and Debreu [1982]' characterizations of ex­

cess demand functions are useful in studying relationships between general

proofs of existence of competitive equilibrium for exchange economies and

fixed-point theorems.

Uzawa [1962] proved that Brouwer's Fixed-Point Theorem is equivalent

to the following existence theorem:

Statement A (cf. Gale [1955], Nikaido [1956] & Debreu

[1956]). For every excess demand function ( : D. -t JR!, there

exists an equilibrium p E D., i.e. ((p) ~ O.

6

However, Uzawa's result only shows that the existence of fixed-points is

equivalent to the existence of equilibrium for excess demand functions de­

fined on t::.. But how does equilibrium for excess demand functions defined

on t::. relate to equilibrium for economies? This has remained an open ques­

tion, which the existing literature (e.g. the Sonnenschein-Mantel-Debreu

Theorem, Mas-Colell's Theorem [1977]' etc.) does not answer because its o

excess demand functions are defined only on certain subsets of t::., not on

t::.. However, our Theorem 1 does cover the situation.

Theorem 3 (Existence of Fixed-Points and Equilibrium Existence).

i) Brouwer's Fixed-Point Theorem is equivalent to Statement A.

ii) Statement A is implied by the Arrow-Debreu Equilibrium Existence

Theorem ([1954], Thm. I).

Proof. Assertion (i) was shown by Uzawa [1962]. And (ii) follows imme-

diately from Theorem 1. Q.E.D.

Therefore, by combining (i) and (ii) in Theorem 3, we see that Brouwer's

Theorem is implied by the Arrow-Debreu Theorem.

3. Technical details.

Here we will establish the two results (Lemma 1 and Proposition 1)

which we have used in the proof of Theorem 1. The following Lemma 1

extends demands from "small" to large price sets, permitting us to focus

on the special case where an excess demand function <: is defined on all of

IR~ \{o}.

7

Lemma 1 (Extension of Excess Demand Functions). Let /{ be a

non-empty compact subset of JR~ \{O}, and ( : /{ -+ JRI be an excess de­

mand function. Then there exists an excess demand function ( : JR~ \ {O} -+

JRI such that (IK = (.

Proof. By (H), it clearly suffices to consider the case where /{ ~ .6.. By

Tietze's Extension Theorem (cf. Munkres [1975], p. 212), there exists a

continuous function ( : .6. -+ JRI such that (IK = (. Then we define a

function ( : .6. -+ JRI by (p) = (p) - (p . (p)lp . p)p. It is clear that (

is an excess demand function such that (IK = (. Finally, we extend the - - - I

function ( to the whole space JR~ \{O} by defining (p) = (pi Ei=l pt) for

any nonnegative nonzero p fI. .6.. This extension is clearly an excess demand

function whose restriction to /{ is (. Q.E.D.

By Lemma 1, the following proposition then establishes the necessity

part of Theorem 1.

Proposition 1 (Characterization of Excess Demand Functions de­

fined on JR~ \{O}). Let ( : JR~ \{O} -+ JRI be an excess demand function.

Then there is an exchange economy C = {(ti,wi, JR~)~=d satisfying (1)

and such that (2) holds for every p E JR~ \{O}.

To characterize excess demand functions on compact subsets of posi­

tive prices, Debreu [1974] used "orthogonal" projection to obtain individual

preferences ti and endowments Wi. By contrast, in order to permit bound­

ary prices p (i.e. some Pi = 0), we will use the "oblique" projection as in

McFadden et al. [1974]. Thus, we will normalize the price set JR~ \{O} to

the set P, the intersection of JR~ with the sphere ofradius 2 IIqll centered

at -q, i.e.

P = {pE JR~: (p+q). (p+q) = 4q .q},

8

(4)

where q is given in (BB). In particular, we will decompose any given excess

demand function ( on P into individual "oblique" excess demand functions

¢1, ... , ¢I : P -t IRJ. By a function t.p : P -t IRJ being an oblique excess

demand junction,9 we mean that there exist a vector a » 0 E JRI and a

continuous function (3 : P -t JR such that:

a) (3(p) > 0, b) t.p(p) = (3(p)h(p),

(5)

for every pEP, where h(p) is the oblique projection along the direction of

p + q of a on the tangent space of p (see. Figure 1 below), i.e. 10

p·a h(p)=a- ( )(p+q).

p. p+q (6)

Then we can apply the rationalization result for oblique excess demand

functions given below in Lemma 3.

Proof of Proposition 1. Let ( : JR~ \ {O} -t IRJ be an excess demand

function. We pick any vector q » 0 given in (BB), and consider the set

P defined in (4). Then we can pick vectors a1,···,al »0 E JRI such

that {a1,···, an} is linearly independent and such that for every pEP,

((p) + p + q is contained in the interior of the convex cone spanned by

{a1,···,at} (see Figure 1 below, cf. also Mas-Colell [1977, p.125].) For

every pEP, we can write ((p) + p+ q = E~=l {3i(p)ai, where (3i(p) > 0, in

a unique fashion. It is clear that the functions (3i(p) are continuous. It is

9 This notion is a modification of the function defined in McFadden et. a!. [1974], p. 364, Equation (2). There, is defined only on the relative interior of P, {3 is a not­necessarily-continuous function, and a is only required to be nonnegative nonzero (i.e. not necessarily positive). Also, when q = 0 and a = ei the positive unit vector on the i coordinate axis of R', then h(p) given in (6) becomes Debreu's orthogonal projection function bi(p) = ei - ((p. e;)/(p. p»p.

10 Actually, h(p) is well-defined for every pER' with (p + q) . (p + q) = 4q . q, by the following fact: *) (p + q) . (p + q) = 4q . q implies p . (p + q) > o. To see (*), consider any such a p. Then we have: p. (p + q) = (q + q) . (q + q) - q . (p + q) = (q + q) . (q + q) - (q + q) . (p + q) + q . (p + q) ~ q . (p + q). Therefore, p . (p + q) ~ (1/2)(p. (p + q) + q . (p + q» = (1/2)(p + q) . (p + q) = 2q· q > o.

9

also clear that these ai can be chosen so that for every i, conditions (A-B)

in Lemma 3 below holds with a = ai.

Then consider the oblique excess demand functions tPi : P -+ IR} de­

fined by tPi(p) = f3i(p)hi(p), where hi(p) is defined in (6) with a = ai (see

Figure 1 below). By Lemma 3 below, there exists an exchange economy

£ = {(~i,Wi,m~)~=d satisfying (1) and such that for every i = 1,·· ·,1

and every E P, equation (8) below holds with r.p = tPi, W = Wi and ~=~i.

Also, by an easy linear algebra argument (cf. McFadden et al. [1974], p.

365), it follows that ((p) = E~=l tPi(p) for all pEP, and so (2) holds for

very pEP. Then by (H), we have: (2) holds for every p E m~ \{o}. Q.E.D.

We now show, as claimed in our proof of Lemma 2, that an appropriate

choice of the oblique projection parameters q and a yields an endowment

and preference relation rationalizing individual oblique excess demands on

P.

Lemma 3 (Rationalization of Individual Oblique Excess Demand

Functions on P). Let q, a:» 0 Em', and define P by (4). Let f3 : P -+ m be a continuous function satisfying (5. a). Let r.p : P -+ m' be the oblique

excess demand function defined by (5.b). Assume:

A) a·a=4q·q, and

B) a-qftm~.

Then there exist a vector W :» 0 E m' and a preference relation ~ on m~

such that

~ is continuous, strictly convex and insatiable (indeed :c + Aa ~ :c for every A > 0 E m and every :c E m~),

(7)

and

{r.p(p)} = {:c E m~ : [p.:c $ p. w]

& ('v'y E R+)[(p. y $ p ·w)::} (:c ~ y)]} - {w} (8)

10

for every pEP.

Lemma 3 requires an (insatiable) rationalization ~ on all of P. Debreu

[1974] showed how to construct (monotone) preferences rationalizing his

individual "orthogonal" excess demand functions on compact subsets Sf of

S. To apply the spirit of Debreu to our context, we will prove the following

Lemma 4, where we obtain rationalizations on compact subsets, but of a

larger space

Q = {p E JR' : (p + q) . (p + q) = 4q . q and p . a ~ O}, (9)

which contains all of P (see Fig. 2 below). (Cf. Footnotes 15 and 18.)

Also, there we extend the consumption space from JR~ to JR' , and construct

(insatiable) preferences defined on JR' , not just JR~. As we will see, Lemma

3 follows as a consequence of Lemma 4.

Lemma 4 (Rationalization of Individual Oblique "Excess Demand

Functions" on Qf). Let q, a » 0 E JR', and define Q by (9). Let (} =

maXpeQ p . a. Let f3 : Q -t JR be a continuous function satisfying (5.a) on

Q. Let cp : Q -t JR be the corresponding oblique "excess demand" function,

i.e. cp is defined by (5.b).11 Let f > 0 E JR and define

Qf = {p E Q : f ~ p. a ~ (} -fl. (10)

Then there exist a preference relation ~ on JR' such that:

i) ~ is continuous and strictly convex, (11) ii) z + Aa ~ z for every A> 0 E JR and every z E JR',

and

{cp{p)} = {z E Bp : z ~ y for every y E Bp} (12)

11 By Footnote 10, h(.) is well-defined on Q, and so does rp(.).

11

for every p E Qf' where Bp = {x E JRl : p' x ~ OJ.

Before proving Lemma 4, we apply it to prove Lemma 3.

Proof of Lemma 3. Since Lemma 4 obtains a preference relation only

for sets Qf' we first ensure that P is contained in some Qf' By an easy

geometry argument, it is clear that (A) implies that the vector a maximizes

uniquely the function p ~ p . a on the sphere {p E JRl : p . p = 4q . q}, and

therefore the vector p = a - q maximizes uniquely the function p ~ p' a on

Q. By assumption (B), p ft P, and hence: p. a < p. a = ci for every pEP.

Also, since a » 0 and P ~ JR~ \{O}, we have: 0 < p. a for every pEP.

Then it follows that P ~ Qf for every sufficiently small f > O.

Now, let f3 : P -t JR and cp : P -t JRl be as given in Lemma 3. It is

clear that we can pick a continuous function P : Q -t JR such that Pip = f3

and P(p) > 0 for every p E Q. We define a function cp : Q -t JRl by

cp(p) = P(p)h(p), and we pick any f > 0 with P ~ Qf' By Lemma 4,

there exists a preference relation t on JRl satisfying (11) and such that

{cp(p)} = {x E Bp : xty for every y E Bp} for every p E Qf' and hence

{cp(p)} = {x E Bp : xty for every y E Bp} for every pEP. Then we pick

any w » 0 E JRl with cp(P) + w » 0, and it is clear that the translation !:

of t from JRl to JR~ defined by x !: y ¢::::> x - w t y - w satisfies (7)

and such that (8) holds for every pEP. Q.E.D.

It remains to prove Lemma 4. As mentioned earlier, our proof follows

the spirit of Debreu's proof [1974]: our steps (stages (3.1-7) below) parallel

his. However, we require modifications so that we can rationalize oblique

excess demand functions (rather than orthogonal ones) (cf. Footnote 14)

and allow boundary prices or even negative prices (cf. Footnote 19).

As in Debreu [1974], we endow the set C of non-empty, closed subsets

12

of JRI with the topology of closed convergence (see Hildenbrand [1974],

p.15). From here on, continuity of a function from any subset of JR into C­

is always understood with respect to that topology.12

To illustrate the ideas of our proof of Lemma 4, consider the following

facts 13:

Facts (Indirect Rationalization). For every p, r E Q:

a) p. h(p) = O. b) iEr -I p and 0 < r· a ~ p. a, then r· h(p) > O.

(13)

Proof. Fact (13.a), which states that h(p) is in the tangent space of p,

holds as h being a projection function. To prove Fact (13.b), we apply the

arguments in McFadden et al. [1974], p.364, as follows. Consider any pair

of distinct r,p E Q. By definition (6), we have:

r·h(p) = r.a-p.a(r.(p+q)/p·(p+q)). By (*) in Footnote 10, we have:

p.(p+q) > O. Also, r.(p+q)-p·(p+q) = (r+q)·(p+q)-(p+q).(p+q) < O.

Therefore, we have: r· h(p) > 0, and so (13.b) holds.

Q.E.D.

Fact (13.b) suggests that the function p I-t -p . a plays a role similar

to an indirect utility function for the oblique projection h.14 For example,

by viewing -r' a and -p' a as the indirect utilities of rand p, and h(p) as

the bundle purchased under p, one can interpret (13.b) as follows: if p-lr

and the indirect utility at r is less than or equal to that at p, then what

12 For any X ~ n ', the interior of X is denoted by IntX, and the boundary of X is denoted by ax.

13 Fact (13.b) has its antecedent in McFadden et al. [1974], p. 364. There the price set is the relative interior of P, rather than Q, and a is any nonnegative nonzero vector (i.e. not necessarily positive).

14 By contrast, Debreu [1974] used p .... -p . e, as the indirect utility function for his orthogonal projection function b'(p) = e, - «p. e,)/(p. p»p, where e, is the i-unit vector.

13

is purchased under p is not affordable at r.15 Similar intuition can also be

applied to the function <.p = f3h as given in the Lemma 4, where f3 > 0 (as

in (5.a)).

Corresponding to the "indirect utility" function p I-t -p. a, we will

define the indirect weakly-preferred sets Ut = {p E Q : p . a :::; t}, the

indirect indifference sets Vt = {p E Q : p. a = t}, and the indirect weakly­

worsen sets L t = {p E Q : p . a ~ t}, where t E [0, a]. (See Figure 2

below.) Actually, in our proof of Lemma 4, we will transform the sets Ut

into a profile of (direct) weakly-preferred sets M t which constitutes to a

preference on JR' satisfying (11) and rationalizing <.p (i.e. (12) holds) on Q£.

Each (direct) weakly-preferred set M t we will construct is correspond­

ing to the indirect weakly-preferred set Ut and rationalizes <.p, therefore it

must satisfy: i) Mt is weakly above the budget set Bp for every pELt, and

ii) for every p E Vt, the set Mt intersects with Bp only at <.p(p). This moti­

vates our use of the convex16 cone L; = {x E JR' : p . x ~ 0 for every p E

L t }, which is the set of all commodity bundles which are either unaffordable

or just-affordable at every pELt. Thus (i) simply states: i') M t ~ L;.

Also, it can be verified from (14) below and that for every t E (0, a) and

p E Vt, the convex cone L; (which by definition is weakly above Bp) in­

tersects with Bp only at the ray {Ah(p) : A ~ a}. Therefore, (ii) simply

means: ii') Mt n In; = <.p(Vt). (See Figure 2 below; ref. (15.a) below; cf.

Debreu [1974], p.18, paragraph 3.)

15 This intuition fails at anypairofboundarypointsp,r ofQ, wherep·a = r·a = O. In particular, for any such a pair of p, r, we have: h(p) = h(r) = a, and r.h(p) = p.h(r) = O. By this fact, and by picking {3 with {3(p) #: {3(r), one can construct an oblique "excess demand" function iP violating the Weak Axiom of Revealed Preference at P. r, and so it is impossible to rationalize iP on the whole set Q by any preference t. This explains why in Lemma 4, we can obtain rationalization only on Q., rather than the whole set Q.

16 By (13) and (14) below, it can be verified that L: is indeed a strictly convex cone (i.e. any ray in 8L: passes through 0) for every t e (0,&). This observation, however, will not be used in our proof of Lemma 4. because we will use (13) and (14) directly.

14

We now characterize the interior and boundary of L;. This result is

useful for constructing the sets M t .

Facts (Characterization of IntL; and (n;). a) For every t E [0, a],

IntL; = {x E JRI : p . x > 0 for all PELt}.

b) For every t E (0, &) and every x E JRI,

i) if x = 0, then x E 8L;, and

(14.a)

ii) if x :j:. 0, then one has: x E 8L; if and only if (14.b) x = )"h(p) for some).. > 0 and some p E vt.

Proof. To show part (a), consider any t E [0, &] and any x E R. It is easily

verified that if p . x > 0 for every pELt, then x E IntL;. Conversely, let

x E IntL;. Suppose p. x = 0 for some pELt; then for any sufficiently small

u> 0, we have: x - up E IntL; ~ L; and p. (x - up) = -up· p < 0; and

a contradiction is derived. Thus we must have: p. x> 0 for every PELt.

And (14.a) holds.

To show part (b), it is clear that (14.b.i) is immediate from (14.a). It

remains to show (14.b.ii) for any t E (0, &) and any nonzero x E JRI . First,

suppose x = )"h(p) for some).. > 0 E JR and some p E vt ~ Lt. Notice that

for every r E Lt , we have: r· a? t = p. a, and so r· x = )..r . h(p) ? 0 by

(13). Therefore, x E L;. Then by (13.a) and (14.a), we have: x E 8L;.

Now, suppose x E 8L;. By (14.a), there exists apE Lt with p. x = O.

We will show that: A) p E vt and B) there exists).. > 0 with x = )"h(p).

To show (A), we suppose by contradiction p rt vt, i.e. p. a > t. Then

consider the function F(-y, u) = (-yp - ux + q) . (-yp - ux + q) - 4q· q, which

has F(l,O) = 0, and 8F~~,O) = 2p· (p+ q) > 0 by fact (*) in Footnote 10.

By the Implicit Function Theorem (cf. Rudin [1976], p. 224), we can pick

a iT > 0 small enough and a l' close enough to 1 so that F(1', iT) = 0 and

15

(.:yp - ux) . a > t. Then we have the contradiction .:yp - UX E Lt and

(.:yp - ux) . x = -ux . x < O. Thus have shown (A).

To show (B), observe that p . x = 0 :::; p . x for every pELt. Then p

minimizes p. x subject to: i) p. a = t and ii) (p+q). (p+q) = 4q· q. The first

order condition (cf. EI-Hodiri [1970], p.27) of this minimization problem

yields A, p E IR with x = Aa+2p(p+q). Since 0 = p·x = Ap·a+2pp·(p+q) =

0, we have: x = A(a - (p. alp· (p + q))(p + q)) = Ah(p). It remains to

show that A > O. First, since x :I 0, we have A :I o. Second, we suppose

by contradiction A < 0, then we can pick any r :I pin L t , and so by (13.b)

we have: r· x = Ar· h(p) < 0, contradicting to the fact that x E L;. Thus

we have A > 0, and hence (B) holds. This completes our proof for (14.b).

Q.E.D.

We now give several useful topological properties for the cones L;. First, for every t E [0, a], since L t is compact, it follows that L; is closed.

Then by the Krein-Milman (cf. Royden [1988], p. 242), it follows from (14)

that L; is indeed the convex cone spanned by h(Vi). Also, it is clear that

each Vi is compact; and it is easily verified (e.g. using the Implicited Func­

tion Theorem) that the function t ~ Vi is continuous on [0, a]. Therefore,

the function t ~ L; continuous on (0, &) into £.17

We now present our proof of Lemma 4.

Proof of Lemma 4. Let a, q, Q, &, f3 and <p be as given in Lemma 4.

Consider any (: > O. Notice that the claim of the lemma is trivial when

Qf is empty. We now consider the non-trivial case, i.e. Qf is non-empty.

We will construct a family {Mt : t E (-oo,oo)} of (direct) weakly-preferred

sets (defined in (25) and (27) below) which satisfies the following properties

17 Also, by using (13) and (14), it can be shown the following hereditary property: if t' < t, then L;, \{O} C IntL;. This property, however, is not required in our proof of

Lemma 4.

16

(see Fig. 2 below):

a) (Rationality) for every t E [f, Q - f]: i) M t ~ L;,

ii) M t n 8L; = <p(lIt), b) (Closedness) every M t is closed, c) (Graduality) the function t t-+ M t is continuous, d) (Open Hereditary) ift' < t, then Mt' ~ IntMt, (15) e) (Strict Convexity) every M t is strictly convex, f) (Insatiability) IntMt 3 x + Aa for every A > 0 E JR,

every x E M t , and every M t ,

g) (Progressiveness) for every x E JRI , there exist t,t' E (-00,00) with x E Mt and x ¢ Mt,.

Before constructing these M t , we define a preference relation t on JRI by

x t y ¢::::::> (Vi E (-oo,oo))[y E Mt => x E Mt]. By (15.b-c) and (15.g), it

is easily verified that the preference t has {Mt : t E (-00, oo)} as its family

of weakly-preferred sets, {8Mt : t E (-oo,oo)} as its family of indifferent

sets, and {IntMt : t E (-oo,oo)} as its family of strictly-preferred sets.

Then it follows clearly from (15.b-g) that t satisfies (11).

We now show that t rationalizes <p on QE' i.e. (12) holds for every

p E QE' Consider any p E QE' and let t = p. a. We need to show that:

A) Mt' n Bp = 0 for every t' < t, and B) Mt n Bp = {<p(p)}. To see (A),

consider any t' < t. By (15.a.i) and (15.d), we have: Mt' S; IntMt S; IntL;,

and so by (14.a), we have: p.y > 0 (hence y ¢ Bp) for every y E Mt,. Hence

(A) holds. To see (B), by (15.a.ii), (15.d) and (14.a) we have: p . y > 0 for

every y E M t \<p(lIt). Also, by (13.b) and (5.b), we have: p. y > 0 for every

y = <p(p') with p =P p' E lit. Then we have: M t n Bp S; {<p(p)}. Also, since

p E lit, by (15.a.ii) we have <p(p) S; M t . Notice that <p(p) E Bp. Thus we

have: M t n Bp ;2 {<p(p)}. Hence (B) follows. And so (12) holds for every

pE QE'

As mentioned earlier, we will obtain the profile of weakly-preferred

sets M t from the profile of indirect weakly-preferred sets Ut by making use

17

of the sets L;. This will be done along the lines of Debreu [1974], with

appropriate modifications (e.g. see Footnote 19).

(3.1 Indirect Stage: picking Ut , where t E [E, a- E] ) We begin by listing the

following properties for the family {Ut : t E [E, a - En of indirect weakly­

preferred sets:

a) (Compactness) every Ut is compact, b) (Graduality) the function t t-t Ut is continuous, (16) c) (Hereditary) if t' < t, then Uti CUt,

Facts (16.a) and (16.c) are obvious. And (16.b) is easily verified (e.g. by

using the Implicited Function Theorem).18

(3.2 Direct Stage: transforming Ut to Dt , where t E [E, a-E]) To transform

the indirect preference to a direct preference, as the first step we define

Dt = tp(Ut) for every t E [E, a - E] (see Figure 3 below). Then the sets Dt

satisfy the following properties:

a) (Rationality) for every t, i) D t ~ L;,

ii) Dt n En; = tp(Vt), b) (Compactness) every Dt is compact, c) (Graduality) the function t t-t Dt is continuous, d) (Hereditary) if t' < t, then:

i) Dt l ~ Dt , and ii) Dt' ~ IntL;.

(17)

Facts (17.b-c) are clear from (16.a-b). And (17.d.i) holds by (16.c). To

prove the rest of them, consider any t E [E, a - E] and any x E Dt = tp(Ut}.

Then x = {3(p)h(p) for some p E Ut . To see (17.a.i), for every r E Lt , since

r·a ~ t ~ p·a, it follows from (13) that r·x = {3(p)r.h(p) ~ O. Thus, x E L;,

and so (17.a.i) holds. We now show (17.d.ii). Suppose x E Dtl for some

t' < t. By repicking p, we can assume p E Uti, and so p. a = t' < t :$ r· a

18 By contrast, the graduality property is not necessary for the profile of sets Ut nP. This explains why we have to extend our price set from P to Q.

18

for every r E Lt. By (14.b), we have: r· x = f3(p)r . h(p) > 0 for every

r E L t , and so x E IntL; by (14.a). Thus (17.d.ii) holds. It remains to show

(17.a.ii). Suppose p E Vt. Then p. x = f3(p)p. h(p) = 0, and so: x E (n; by (14.b). Thus, we have: Dt n 8L; ;2 tp(Vt). Also it follows from (17.d.ii)

that D t n8L; ~ tp(Vt). Therefore (17.a.ii) follows.

(3.3 Open Hereditary Stage: transforming Dt to E t where t E [f, a-f].) The

sets Dt satisfy only the property of hereditary, but not the open hereditary.

To obtain this property, we will expand each Dt to a union of closed balls

Nt•x centered at x EDt, while maintaining rationality, graduality and

compactness. (See Figure 4 below.)

In defining the radii of the closed balls Nt •x , we will make use of the

function d : [E, a - f] X JRI --t JR defined by d(t, x) = min{llx - yll : y E

UPEL,Bp}. Since the sets L t are compact, it follows that the sets UPEL,Bp

are closed, and so the function d is well-defined. Also, it is easy to see that

d(t, x) is non-increasing in t. Moreover, observe that each UPEL,Bp is the

closure of the complement of L;, then by the continuity of the mapping

t I-t L;, it follows that the function d is continuous.

We now define the closed ball Nt •x = {y E JRI : IIy-xll::; (tja)d(t,x)}

for every x E D t and every t E [f, a- fl. We define the set Et = UXED,Nt.x

for every t E [f, a - fl. Then the sets E t satisfy:

a) (Rationality) for every t, i) E t ~ L;,

ii) E t n 8L; = tp(Vt), b) (Compactness) every E t is compact, c) (Graduality) the function t I-t E t is continuous, d) (Open Hereditary) if t' < t', then EtA ~ IntEt .

(18)

Facts (18.b-c) follows clearly from (17.b-c) and the continuity of the func­

tion d. To show (18.d), let t' < t. For any x EDt', we have: x E Dt

by (17.d.i). Then it clearly suffices to show that Nt ,.x ~ IntNt •x . By

19

(17.d.ii) we have 0 < d(t, z). Since d(t', x) ~ d(t, x) and t' < t, we have: t' t --{;d(t', x) < {;d(t, x). Therefore, we have: Nt"x ~ IntNt,x; and so (18.d)

holds. To show (18.a), consider any t and x EDt. It is clear that Nt,x ~ L~,

and so E t = UxeD,Nt,x ~ Lt, i.e. (18.a.i) holds. It remains to show

(18.a.ii). Since Et ;2 Dt , by (17.a.ii) we have: Et n 8L; ;2 tp(Vt). We now

show that Et n 8L; ~ tp(Vt). To see this, consider any y E Et n 8L;. Then

y E Nt,x for some x EDt. Suppose by contradiction that x E IntL;. Then

we clearly would have: Nt,x ~ IntL;, and so: IntL; ;2 Nt,x :3 Y E 8L;, a

contradiction. Therefore, we must have: x E 8L;; so d(t, x) = 0, and hence:

Nt,x = {x}. Therefore, we have: 8L; :3 Y = x EDt; and so y E tp(Vt) by

(17.a.ii). Thus, we have: E t n 8L; ~ tp(Vt). Hence (18.a.ii) holds.

(3.4 Convexity Stage: transforming Et to Ft , where t E [f,o- - f].) Since

Et may not be convex, we let Ft be the convex hull of Et . (See Figure 5

below.) Then the sets Ft satisfy:

a) (Rationality) for every t, (i) Ft ~ L;,

(ii) Ft n 8L; = tp(Vt), b) (Compactness) every Ft is compact, c) (Graduality) the function t t-t Ft is continuous, d) (Open hereditary) if t' < t, then Ft , ~ IntFt, e) (Convexity) every Ft is convex.

(19)

Facts (19.b-d) follow clearly from (18.b-d). And (19.a.i) follows from (18.a.i)

and the fact that the cones L; are convex. To see (19.a.ii), consider any

t E [f,o- - fl· Since Ft ;2 E t , by (18.a.ii) we have: Ft n 8L; ;2 tp{Vt).

To show that Ft n 8L; ~ tp{Vt), consider any x E Ft n 8L;. By the

definition of Ft , we have: x = L:~=l AiXi for some n, some real numbers

AI, ... ,An > 0 with L:~=l Ai, and some vectors Xl, ... Xn E E t ~ L;. Since

x E 8L;, we have: Xl,"', Xn E 8L;. By (18.a.ii), there exist PI,'" ,Pn E Vt

such that Xi = tp{Pi) for all i. We claim that PI = ... = Pn. Suppose

not, then it follows from (13.b) that for every r E L;, we have: 0 <

20

f3i(p;)r . h(Pi) = r . IP(Pi) = r . Xi for some i. Also, for every i, since

Xi E L~, we have: r· Xi 2:': O. Therefore, we have: r· X = 2:7=1 Air· Xi > 0

for every r E L t ; and so X E IntL~ by (14.a). Thus we have derived the

contradiction: IntL~ 3 X E 8Lt. Therefore, we must have: P1 = ... = Pn;

and so x = Xl = '" = Xn E IP(Vt). Hence Ft n 8L; 2 IP(Vt). And (19.a.ii)

holds.

(9.5 Insatiability Stage: transforming Ft to Gt , where t E [e, it - e].) Since

the sets Ft does not satisfy the instability property. To obtain this prop­

erty, we will adding them with a convex cone A spanned by a closed ball

centered at a, while maintaining the properties of rationality, graduality

and convexity.19 (See Figure 6 below.) Of course, this leads us to weaken

the property of compactness to closedness.

The closed ball we pick can be any No = {x E JRI : IIx - all < c5} with

c5 > 0 E JR and such that p. x > 0 for every pELf and every x E No. We

then define the cone A = {AX: A 2:': 0 E JR, x E No}, which is clearly closed

and convex, and has non-empty interior. Also, for any z E A, any pELt

and any t E [e, it - e], we have: i) p. z 2:': 0, and ii) p. z > 0 if z # O. Then

by (i) we have: A ~ L;; and by (ii) we have: z E IntL; for every non-zero

z E A. Also, it is easy to see that z + Aa E IntA for every z E A and every

A> 0 E JR.

For every t E [C, it - e], we define Gt = Ft + A. Then the sets Gt satisfy

19 Debreu [1974] made use of n~ to obtain monotone weak-preferred sets. Our

convex cone A shall be a proper subset of n' in order to maintain rationality property for allowing "non-positive" prices P (i.e. Pi < 0 for some i) in L 1• (Cf. the proof of (20.a.i) given below.)

21

the following properties:

a) (Rationality) for every t, i) Gt ~ L;,

ii) G t n 8L; = <p(Vt), b) (Closedness) every Gt is closed, c) (Graduality) the function t t-+ Gt is continuous, (20) d) (Open Hereditary) if t' < t, then G t , ~ IntGt ,

e) (Convexity) every Gt is convex, f) (Instability) x + Aa E IntGt for every A > 0,

every x E Gt and every Gt .

Fact (20.b) holds because of (19.b) and the fact that A is closed. Facts

(20.c-d) follow clearly from (19.c-d). Fact (20.e) follows clearly from (19.e)

and the fact that A is convex. Fact (20.a.i) follows from the fact that

each L; is a convex cone containing Et and A. It remains to show (20.f)

and (20.a.ii). Consider any t and any x E Gt . Then x = y + z for some

y E Ft and some z E A. To see (20.f), consider any A > O. Then we

have: z + Aa E IntA, and so x + Aa = y + z + Aa E Ft + IntA ~ IntGt .

Thus (20.f) holds. To see (20.a.ii), since Gt ;2 Ft , by (19.a.ii) we have:

G t n 8L; ;2 <p(Vt).We now show that G t n 8L; ~ <p(Vt). Suppose x E 8L;.

Then we have: z = 0, because otherwise, we would have z E IntL;, and

so x = y + z E IntL;. Therefore, we have: Ft 3 y = x E 8L;. Then by

(19.a.ii), we have: x E <p(Vt). Thus, we have shown that G t n 8L; ~ <p(Vt).

Hence (20.a.ii) holds.

(3.6 Strict Convexity Stage: trons/orming Gt to Mt , where t E [f,o.­

f].) The sets Gt are not strictly convex. To obtain this property, we will

transform them to the sets M t as given in the beginning of this proof of

the lemma, where t E [f, 0.- fl. To do this, we pick any vector p E Q with

p·a = a and let Tp be the tangent space ofp, i.e. Tp = {y E JRI : p.y = OJ.

We will define strictly concave functions JJt : Tp -t JR, and will define

Mt = {y + sa : y E Tp and JJt(Y) $ s E JR}. (See Figure 7 below, cf. also

(26) below.)

22

To define the functions f..lt, we first define two functions as follows. For

every t E [e, 0- eJ and every y E Tp, let It(Y) be the least s such that

y + sa E Ct , and let At (y) be the least s such that y + sa E L;. (See Figure

7 below.) We claim: i) 0 ~ At{y), ii) At(Y) ~ It{y), and iii) It{y) < 00. To

see (iii), it suffices to find any s with y + sa E Ct. To do this, we can pick

any x E Ft and pick any s E JR with y - x E s(Na - a), where Na is given

in the second paragraph in Stage 3.5. Then we have: y + sa = x + sz for

some z E Na , and so y+sa E Ft+A = Ct. Hen<;e (iii) holds. To see (i), by

(iii) we have: y + sa E Ct ~ L; for some s E JR. Also, notice that for every

s < 0 we have: p. (y + sa) = sp· a = so < 0, and so: y + sa f/. L; because

pELt. Therefore (i) follows. Finally, (ii) is clear from (20.a.i). Thus the

claim is established.

We now list four useful properties for the functions (t, y) ~ At(Y) and

(t, y) ~ It(Y). First, they are continuos, by (20.c) and by the continuity

of the mapping t ~ L;. Second, each At (.) and It (.) is a convex function

on Tp for every t, because each Ct and L; is a convex set. Third, for every

t and every y E Tp, since the sets L; and C t are closed, it follows that

y + At (y) E en; and y + It (y) E aCt. Fourth, for every t and every y E Tp,

by the previous property and (20.a.ii), we have: At(Y) = IdY) if and only

if Y + At(Y) E <p(Vt).

We pick a continuous convex function p defined on {(s, s) E JR~ : s ~

s} (e.g. the one given in the appendix of Debreu [1974]) such that:

(a) p is strictly increasing in each variable, (b) p(s, s) = s for every s E JR+, (21) (c) if s -:/= sand/or s' -:/= s', then p is strictly convex on

the segment [(s, s), (s', s')].

Then we define JJt (y) = p( At (y), It (y)) for every t E [e, a] and every Y E Tp.

Clearly, the function (t, y) ~ JJt(Y) is continuous.

We now show that JJt(-) is a strictly convex function for every t E [£, a].

23

To see this, consider any t E [f, a - f], any pair of distinct vectors y, y' E Tp,

and any r E (0,1). Then:

Jlt(ry+ (1- r)y')

= p(>'dry + (1- r)y'), 'Yt(ry + (1- r)y'»

~ p(r>.t{y) + (1- r)>'t(Y'), T'Yt(Y) + (1 - rht(Y')) (22)

~ rp(>.t{y), 'Yt (y» + (1- r)p(>'dY'),'Yt(y'» (23)

= rJ-lt(y) + (1- r)J-lt(Y')·

Therefore, it suffices to show that either (22) or (23) is a strict inequality.

There are three possible cases, as follows.

First, suppose >'t(Y) = >'t(y'). Notice that y + >'t(y)a, y' + >'t(y')a E

8L;. By (14.b), there exist real numbers 8,8' ~ 0 and vectors p,p' E \It

such that y + >'t(z)a = 8h(p) and y' + >'t(z')a = 8'h(p'). We claim that

p :/; p'. Suppose not. Then we have: (8 - 8')h(p) = (y - y'), and hence:

(8 - 8')p . h(p) = p. (y - y') = O. Notice that 8 :/; 8' because y :/; y'.

Therefore we have: p. h(p) = 0, contradicting to (13.b) because a . p =

a > t = p. a. Thus we must have: p :/; p'. Similarly, it can be shown

that 8,8' > O. Then it follows from (13.b) that for every pELt, we have:

o < p·(r8h{p)+(1-r)8' h(p'» = p.{ry+(1-r)y' +(r>'t(z)+(1-r)>'t(y'))a).

Therefore, by (14.a), we have: ry+ {1- r)y' + (r>'t(z) + {1- r)>'t(y'))a E

IntL;. Then it follows from the definition of >'t(ry + (I - r)y') that we

have: >'t(ry + (I - r)y') < r>'t(Y) + (1 - r)>'t(y'). Then by (21.a) and the

convexity of the function 'Yt(-), it follows that the inequality (22) is strict.

Second, suppose >'t(Y) :/; >'t(y'), and suppose >'t{Y) = 'Yt(Y) and >'t(y') =

'Yt(y'). Then there exist p,p' E \It such that: y + >'t(y)a = <p(p) and

y' + >'t(y')a = <p(p'). We claim that p:/; p'. Suppose not. Then we have:

y+>'t{y)a = y' +>'t{y')a; and hence have the contradiction: 0 = p{y-y') =

(>'t(Y) - >'t{y'))p· a = (>'t(Y) - >'t(y'))a :/; O. Thus, we must have: p:/; p'.

24

By applying the arguments used in the last paragraph, it follows that the

inequality (22) is strict.

Third, suppose AdY) i= At(y') and suppose At(Y) i= It{y) and/or

At(y') i= Idy')· Then (At(y)"t{y)) i= (At(Y'), Idy')), and so (21.c) im­

plies that the inequality (23) is strict. Thus we have shown that J.ld·) is a

strictly convex function for every t E [f, Q - f).

As mentioned earlier, we define:

Mt = {y + sa : y E Tp, s ~ J.lt(Y)} for every t E [f, Q - f). (24)

We now show that:

{Mt : t E [f, Q - f]} defined by (24) satisfies (15.a-g). (25)

Properties (15.b-c) are satisfied because Tp is closed and the function (t, y) ...-+

I-'dY) is continuous. Property (15.e) will follow immediately from the fact

that each I-'t(-) is strictly convex, once we have shown (15.f). Our prooffor

(15.f) requires the following decomposition fact:

each x E JRI has uniquely a vector 7r:c E JRI and a s E JR such that: x = 7r:c + s:ca; in particular, one has: S:c = r.; (26) and 7r:c = X - s:ca.

Thus 7r:c is the projection of x along the direction of a on Tp. To see

(26), since p . a = Q > 0, and so 7r:c and S:c are well-defined. By algebra

calculation, it is easy to see that x = 7r:c + s:ca. To see the uniqueness of

the decomposition, let x = Y + sa, where y E Tp and s E JR. Then have:

p. x = p. y + sp· a = sp· a. Therefore, s = S:c and so y = x - s:ca = 7r:c.

We now show that {Mt : t E [f, Q - f]} satisfies (15.f). Consider any

t E [f, Q - f) and any x E Mt . Let x = y + sa, where y E T£ and s ~ I-'t(Y).

Consider any A> O. We need to show that the vector x, = x + Aa E IntMt .

By (26), x' has a unique decomposition: x' = 7r:c,+s:c,a, where 7r:c, E Tp and

25

sx' E JR are as defined in (26). Since y+(s+>.)a = x+>.a = x' = 1rx,+sx,a,

we have: 1rz' = Y and Sx' = s+>. > I-'t{y) = 1-'t{1r~), and so x' E Mt . We now

show that x' is indeed an interior point of M t . Notice that the mappings

x I-t Si and x I-t I-'t{ 1ri) are continuous on JRI; therefore, we can pick any

open ball N 3 x' such that Si > I-'t (1ri) for every x EN. Then we have:

x' E N = {1ri + Sia : x E N} C {1ri + sa : s ;::: 1-'t{1ri) , and x E N} C

{y + sa : s ;::: I-'t(y), and y E Tp} = Mt . Thus, x' E IntMt . And we have

shown that {Mt : t E [t,o- - t]} satisfies (15.f)

We now show that {Mt : t E [t,o- - t]} satisfies (15.d). Consider any

t,t' E [t,o-- t] with t' < t. Then we have: >'t'(');::: >'to, and "}'t'O > "YtO

by (20.d). By (21.a), we have: I-'t'(·) > I-'t(·). Now consider any x' E Mt"

and let x' = y + sa, where Y E Tp and s ;::: I-'t'(Y)' Then we have: x' = x+(s-I-'t{y))a, where x = Y+I-'t(y)a E Mt· Since S-I-'t{y) > S-l-'t'(Y) ;::: 0,

by (15.f) we have: x' E IntMt . Therefore, {Mt : t E [t,o- - t]} satisfies

(15.d).

It remains to show (15.a). Consider any t E [t,o- - t] and any x E M t .

Let x = y + sa, where Y E Tp and S ;::: I-'t(Y). Since "Yt(Y) ;::: >'t(Y), by

(21.a) we have: I-'t(Y) ;::: >'t(Y); and so S ;::: >'t(Y). Therefore, we have:

p. x = p. (Y+ >'t(y)a) + (s - >'t(Y))P' a;::: 0 for every pELt; and so x E L~.

Thus (15.a.i) holds.

To show (15.a.ii), consider any t E [t,o- - t) and any x E JR1. Let 1rx

and Sx be as given in (26). Then x = 1rx + sxa. It is clear that x E aL; if

and only if Sx = >'t(1rx). We claim that: I) x E cp(vt) implies x E Mt naL~,

and II) x E MtnaL~ implies x E cp(vt). To show (I), suppose x E cp(vt). By

(20.a), we have 1rx + sxa = x E L~ n Gt and so: Sx ;::: >'t(1rx), "Yt(1rx). Then

we have: >'t(1rx) = Sx ;::: "Yt(1rx) ;::: >'t(1rz ), and so: Sz = >'t(1rz ) = "Yt(1rz ). By

(21.b), we have: Sz = I-'t(1rz); and so x = 1rz + sza E Mt . Thus, we have:

x E M t n a L~. Hence (I) holds. To show (II), suppose x E M t n a L~. Since

26

x E Mt , we have: x = y + sa for some y E Tp and s E JR with s 2:: J-!t(Y).

By (26), we have: trx = Y and Sx = s; and so Sx 2:: J-!t(7rx ). Since x E L;, we

have: At(7rx ) = Sx. We assert that id7rx ) = Sx. Suppose not. Then we have

id7rx ) > Sx = At(7rx ); and so we have: J-!t(7rx ) > Ad7rx ) = Sx by (21.a-b).

Therefore, we have the contradiction: Sx 2:: J-!t(7rx ) > Ad7rx ) = Sx. Thus

we must have: it(7rx ) = Sx. Therefore, x = trx + sx a = trx + id7r)a E Ct.

Then we have: x E L; n C t , and so x E tp(Vt) by (20.aii). Thus (II) holds.

Then by (I) and (II), we have: Mt n In; = tp(Vt), and therefore (15.a.ii)

holds.

(3.7 Progressiveness Stage: defining Mt for everyt ¢ [f, a-f].) To complete

the construction of a profile {Mt : t E (-oo,oo)} as given in the beginning

of this proof for Lemma 4, we define

M _ { Mf + (f - t)a for every t E (-00, f), t - M t = M&-f + (a - f - t)a for every t E (a - f, 00), (27).

where Me and M&-f are defined in (24). It is easy to see that the family

{Mt : t E (-oo,oo)} of sets defined by (24) and (27) satisfies (15.a-f).

We now show that it also satisfies (15.g). Consider any Z E JR1, which

by (26) has a unique decomposition z = trx + sxa. By the definitions

(24) and (27), it is easily verified that: I) for every t 2:: a - f, if s 2::

J-!&-f(7rX ) + (a - f) - t, then trx + sa E M t ; and II) for every t' S f, if

s < J-!f(7rX ) + (f - t), then trx + sa ¢ Mt,. Therefore, we have: x E Mt

for every t 2:: max{a - f,J-!&-f(7rX ) + (a - f) - sx}; and z ¢ Mt' for every

t' < min{f,J-!f(7rx ) + f - sx}. Hence (15.g) holds.

Thus we have constructed a family {Mt : t E (-oo,oo)} as claimed in

the beginning of this proof for Lemma 4, and we have shown that it satisfies

(15.a-g).

Q.E.D.

27

References

Arrow, KJ. and Debreu, G. 1954. Existence of an equilibrium for a com­

petitive economy. Econometrica 22: 265-90.

Brouwer, L.E.J. 1910. tiber ein eindeutige, stegie Transformationen von

Flachen in sich. Mathematische Annalen 69: 176-80.

Debreu, G. 1956. Market equilibrium. Proceedings of the National Academy

of Sciences of the U.S.A., 42: 876-878.

Debreu, G. 1974. Excess demand functions. Journal of Mathematical Eco­

nomics,1: 15-2l.

Debreu, G. 1982. Existence of competitive equilibrium. Ch. 15 in Handbook

of Mathematical Economics Vol. 2, edited by KJ. Arrow and M.D.

Intrilligator. Amsterdam: North-Holland.

EI-Hodiri, M. A. 1970. Constrained Extrema: Introduction to the Differen­

tiable Case with Economic Applications. Springer-Velag.

Gale, D. 1955. The law of supply and demand. Mathematica Scandinavica,

3: 155-169.

Hildenbrand, W. 1974. Core and Equilibria of a Large Economy. New

Jersey: Princeton University Press.

Mantel, R. 1974. On the characterization of aggregate excess demand.

Journal of Economic Theory 12: 197-202.

Mantel, R. 1979. Homothetic preferences and community excess demand

functions. Journal of Economic Theory 12: 197-202.

Mas-Colell, A. 1977. On the equilibrium price set of an exchange economy.

Journal of Mathematical Economics 4: 117-126.

Mas-Colell, A. 1985. The Theory of General Economic Equilibrium: A

Differentiable Approach. New York: Cambridge University Press.

McFadden, D., A. Mas-Colell, R. Mantel, and M.K Richter 1974. A char-

acterization of community excess demand functions. Journal of

28

Economic Theory 9: 361-374.

Munkres, J.M. 1975. Topology, A First Course. New Jersey: Prentice

Hall, Inc.

Nikaido, H. 1956. On the classical multilateral exchange problem. Metroe­

conomica, 8: 135-145.

Rodyen, H.L. 1988. Real Analysis. 3rd edition. New York: Macmillan.

Rudin, W. 1976. Principles of Mathematical Analysis. 3rd. edition, 1986.

McGraw Hill.

Sonnenschein, H. 1972. Market excess demand functions. Econometrica,

40: 549-563.

Sonnenschein, H. 1973. Do Walras identity and continuity characterize the

class of community excess demand functions? Journal of Economic

Theory 6: 345-354.

Uzawa, H. 1962. Walras' existence theorem and Brouwer's fixed-point the­

orem. Economic Studies Quarterly, 13: 59-62.

29

I I I I I I I I I I I I

+a1 I I I t I I I I I I

I f3 f.'11 •.. •. " J fJ://aJ. .. · .

. I ..

Figure 1: Oblique Decomposition.

30

.. · .. t;w)+p+q

-L t

Ut

t' < t

BMt

Figure 2: Indirect Preference and Direct Preference.

31

p

.. ~a / ......•

Figure 3: Direct Stage.

32

t' '< t

\ \ \ D' \ t

p

Figure 4: Open Hereditary Stage.

33

p

sa(p)

Figure 5: Convexity Stage.

34

. ----

p

Figure 6: Insatiability Stage.

35 I

y

Tp

I Q) y + At(y)a

o y + rt(Y)a ~ Y+J.lt(v)a

Figure 7: Strict Convexity Stage.

36

.: . . '.

", "

. .